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Moduli Spaces of Pseudo-Holomorphic Disks and Floer Theory of Cleanly Intersecting Immersed Lagrangians

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Moduli Spaces of Pseudo-Holomorphic Disks and Floer Theory of Cleanly Intersecting Immersed Lagrangians MODULI SPACES OF PSEUDO-HOLOMORPHIC DISKS AND FLOER THEORY OF CLEANLY INTERSECTING IMMERSED LAGRANGIANS A DISSERTATION SUBMITTED TO THE DEPARTMENT OF MATHEMATICS AND THE COMMITTEE ON GRADUATE STUDIES OF STANFORD UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY Ken Yin Kwan Chan August 2010 © 2010 by Yin Kwan Chan. All Rights Reserved. Re-distributed by Stanford University under license with the author. This work is licensed under a Creative Commons Attribution- Noncommercial 3.0 United States License. http://creativecommons.org/licenses/by-nc/3.0/us/ This dissertation is online at: http://purl.stanford.edu/bz202yk0512 ii I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. Eleny Ionel, Primary Adviser I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. Yakov Eliashberg I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. Soren Galatius Approved for the Stanford University Committee on Graduate Studies. Patricia J. Gumport, Vice Provost Graduate Education This signature page was generated electronically upon submission of this dissertation in electronic format. An original signed hard copy of the signature page is on file in University Archives. iii Abstract In this thesis we investigate moduli spaces of pseudo-holomorphic disks with La- grangian boundary conditions, in which the Lagrangians are immersed with clean self-intersections. We then discuss the compactification of these moduli spaces, and show that under specific assumptions, the moduli spaces can be oriented. Finally, we use these moduli spaces to construct and compute Lagrangian Floer cohomology for sphere and orientation covers of RPn embedded as a Lagrangian submanifold of CPn. iv Preface There have been tremendous interests in pseudo-holomorphic curves with Lagrangian boundary conditions. For instance, they enter in the definition of Floer's homol- ogy theory of Lagrangian intersections, and constitute the major framework of \La- grangian intersection Floer theory { anomaly and obstruction" [FOOO06], in which Fukaya, Oh, Ohta and Ono laid the foundations for constructing the derived Fukaya Category of a symplectic manifold. The later plays an essential role in Kontsevich's Homological Mirror Symmetry Conjecture. While significant steps were taken to understand the Fukaya Category in Fukaya et. al. [FOOO06] , defining it in complete generality and making rigorous the alge- braic structures involved is a long term and technically involved project. Na¨ıvely, the objects in the category are embedded Lagrangians with extra information. Insofar, morphisms between pairs of transversely intersecting Lagrangians are defined. They are induced from the intersections of the Lagrangians, with algebraic operations de- termined by the information extracted from pseudo-holomorphic discs bounding these Lagrangians. But this only leads to an `precategory' as morphisms are merely defined for pairwise transversely intersecting objects. That said, to obtain a `category', one is led to consider immersed Lagrangians, and allow for intersections other than pairwise transverse ones. Akaho and Joyce took the initial step in [AJ08] to investigate immersed Lagrangians whose self-intersections are transverse double points. This motivates the author to take a further step in extending their work to cover a broader class of immersed Lagrangians, which may possess certain non-transverse self-intersections. It is an on-going project built on the technical machinery of [FOOO06]. v Closely related to the Fukaya Category is the Donaldson-Fukaya Category. It is defined at the (co)-homology level with Floer cohomologies as morphisms, which are associative. This makes it technically more tractable, but at cost of losing information required for Mirror Symmetry after passing to cohomology. Nevertheless, it justifies the attention on Lagrangian Floer cohomology, which can be viewed as one of the by-products of the moduli spaces the author proposes to investigate. vi Acknowledgment First and foremost, I would like to thank my thesis advisor, Eleny Ionel. Over the years, she has been extremely generous with her time, and her advice was always remarkably insightful. I am certainly indebted to her for her patience and her constant encouragement. I would also like to thank Yasha Eliashberg for introducing me to the beautiful subject of symplectic topology, and Sam Lisi for many interesting and helpful discussions. I am deeply appreciative of all those who has given me support and encouragement during my time at Stanford. vii Contents Abstract iv Preface v Acknowledgment vii 1 Introduction 1 1.1 Clean self-intersections . 1 1.2 A local model for clean self-intersections . 3 2 Moduli spaces 5 2.1 Definition of the moduli spaces . 6 2.1.1 Lagrangian boundary conditions . 6 2.1.2 Evaluation maps on Mm+1(J; I; ) ............... 8 2.2 Compactification and boundary . 8 2.2.1 Lagrangian boundary conditions for stable maps . 9 2.2.2 Evaluation maps on Mm+1(J; I; ) . 12 2.3 The linearization . 13 2.4 The virtual dimension . 17 3 Orientability of the moduli spaces 25 3.1 Preliminary definitions . 25 3.2 Orientation of the moduli spaces . 28 viii 4 Lagrangian Floer cohomology 31 4.1 A brief overview . 31 5 Examples and computations 34 5.1 Sphere double covers of RPn in CPn ................... 34 zk 5.2 The moduli spaces M2 ( ) with one jump . 36 5.2.1 Regularity of D@J0 ........................ 37 zk 5.2.2 Orienting the moduli spaces M2 ( j ) . 40 5.3 Morse-Bott Floer cohomology of the sphere covers of RPn . 42 5.3.1 Novikov Ring . 43 5.3.2 Floer coboundary operators . 44 5.3.3 Spectral sequence and Floer cohomology . 47 5.4 Morse-Bott Floer cohomology of the orientation covers of RPn . 55 5.4.1 Floer coboundary operators and M2( ) . 57 6 Further directions 61 6.1 Obstructions to Lagrangian Floer cohomology . 61 6.2 Hamiltonian equivalence . 61 A Spin and relatively spin structures 63 References 66 ix Chapter 1 Introduction The spaces under consideration are a 2n-dimensional symplectic manifold (X; !) with a symplectic structure !, and a smooth n-dimensional manifold L. Throughout this thesis, all manifolds are smooth and without boundary, unless otherwise stated. Sup- pose : L # X is a proper Lagrangian immersion whose image (L) lies in X. By a Lagrangian immersion we mean is a proper immersion with ∗! = 0. The immersion identifies the tangent bundle TL of L with a Lagrangian subbundle of the symplec- tic bundle (∗T X; ∗!). When (X; J; !) is almost K¨ahlerwith a compatible almost complex structure J, one can show that an immersion : L # X is Lagrangian if and only if ∗J maps TL to the normal bundle NL of L in ∗TX. 1.1 Clean self-intersections In [BW97], Bates and Weinstein define the clean intersection of two maps to the same target manifold in terms of their fiber product. We adapt their definition to the case of two identical maps: consider the fiber product, pr2 L×L / L (1.1) FF F prX pr1 FF FF F" L / X 1 CHAPTER 1. INTRODUCTION 2 where pr1 and pr2 are projections on the first and on the second factors respectively. In other words, L×L is the pre-image of the diagonal ΔX :X,! X ×X under the map pr1×pr2. It follows that the fiber product L×L contains a component consisting i of the diagonal ΔL of L . In this and subsequent sections, R will denote the closed ii set (L×L) n ΔL, and K will be the set pr1(R), which is identical to pr2(R) , with an image (K) under . L×L can then be expressed as the disjoint union ΔLqR. If the fiber product L×L happens to be a smooth submanifold of L×L, then for any tangent vector (vx; wy) in T(x;y)(L×L), d(×)(x;y) (vx; wy) is tangent to the diagonal ΔX in X×X, which implies that dx(vx) = dy(wy). But d(pr1)x (vx; wy) = vx, and d(pr1)x (vx; wy) = 0 only when vx = 0. In this case, dx(vx) = dy(wy) = 0. It then follows that the projection pr1 is an immersion. Similarly, pr2 is also an immersion as well. Therefore, we conclude that the projection prX : L×L −! X is itself an immersion if L×L is a smooth submanifold of L×L, in which case the image of L×L under prX is an immersed submanifold of X. This observation motivates the following definition of clean self-intersections. Definition 1.1. An immersion : L # X is said to intersect itself cleanly provided that the fiber product L×L is a smooth submanifold of L×L, and for any point (x; y) in L×L, dx(TxL) \ dy(TyL) = d( ∘ pr1)(x;y)T(x;y) (L×L) (1.2) ∗ as subbundles of ( ∘ pr1) TX. This requires R to be a smooth submanifold of L×L, and consequently L×L is diffeomorphic to LqR. In general, the fiber product L×L may consist of components of possibly different dimensions, which by condition (1.2) are bounded above by the dimension of L. In the special case where -1(p), for all points p in X and a cleanly self-intersecting immersion , has at most two distinct points, the set R, considered i The closeness of (L×L) n ΔL in L×L follows from being an immersion. ii Since there is an involution : L×L ! L×L given by (x; y) = (y; x).
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